D1 June 2010 Q7

EdexcelOld spec11 marksLinear Programming

7.

Figure 6: axes x from 0 to 16 and y from 0 to 24; solid line x = 15 with x > 15 shaded; dotted line y = 6 with y < 6 shaded
Figure 6

Keith organises two types of children’s activity, ‘Sports Mad’ and ‘Circus Fun’.
He needs to determine the number of times each type of activity is to be offered.

Let \(x\) be the number of times he offers the ‘Sports Mad’ activity. Let \(y\) be the number of times he offers the ‘Circus Fun’ activity.

Two constraints are

\[\begin{aligned}x &\leqslant 15\\ \text{and}\quad y &\gt 6\end{aligned}\]

These constraints are shown on the graph in Figure 6, where the rejected regions are shaded out.

(a) Explain why \(y = 6\) is shown as a dotted line. (1)

Two further constraints are

\[\begin{aligned}3x &\geqslant 2y\\ \text{and}\quad 5x + 4y &\geqslant 80\end{aligned}\]
(b) Add two lines and shading to Diagram 1 in the answer book to represent these inequalities. Hence determine the feasible region and label it R. (3)

Each ‘Sports Mad’ activity costs £500.

Each ‘Circus Fun’ activity costs £800.

Keith wishes to minimise the total cost.

(c) Write down the objective function, C, in terms of \(x\) and \(y\). (2)
(d) Use your graph to determine the number of times each type of activity should be offered and the total cost. You must show sufficient working to make your method clear. (5)
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